Ray theory of light predicts that the pattern of a plane wave should maintain the size and shape of the aperture, but diffraction causes light passing through an aperture to deviate from its original direction of propagation. The further the plane wave propagates past the aperture, the more the diffraction pattern changes. The mathematical modeling of diffraction is necessary to understand the image formation properties of an imaging system.
An electromagnetic wave that has traveled to an x-y plane from a parallel ξ-η plane a distance z away can be described by the Huygens-Fresnel Principle (see Introduction to Fourier Optics by Joseph W. Goodman, McGraw-Hill, second edition, 1996) as                               U          ⁡                      (                          x              ,              y                        )                          =                              z                          ⅈ              ⁢                                                           ⁢              λ                                ⁢                                    ∫              ∫                        ∞                    ⁢                                    U              0                        ⁡                          (                              ξ                ,                η                            )                                ⁢                                    ⅇ                              ⅈ                ⁢                                                                   ⁢                                                      2                    ⁢                    π                                    λ                                ⁢                r                                      r                    ⁢                      ⅆ            ξ                    ⁢                                           ⁢                      ⅆ            η                                              (                  Equation          ⁢                                           ⁢          1                )            where U(x,y) is the electric field amplitude in the x-y plane, U0(ξ,η) is the electric field amplitude in the ξ-η plane, λ is the wavelength of the electromagnetic wave, and   r  =                                          (                          x              -              ξ                        )                    2                +                              (                          y              -              η                        )                    2                +                  z          2                      .  Therefore, the Huygens-Fresnel Principle multiplies U0(ξ,η) by a spherical wave before the integration.Fresnel diffraction occurs when the spherical wave can be approximated with a parabolic wave, which is a valid approximation if                     z        >>                              (                                          π                                  4                  ⁢                                                                           ⁢                  λ                                            ⁡                              [                                                                            (                                              x                        -                        ξ                                            )                                        2                                    +                                                            (                                              y                        -                        η                                            )                                        2                                                  ]                                      )                                2            3                                              (                  Equation          ⁢                                           ⁢          2                )            The field amplitude of U(x,y) after propagating a distance z can now be written as                               U          ⁡                      (                          x              ,              y                        )                          =                                            ⅇ                              ⅈ                ⁢                                                                   ⁢                                                      2                    ⁢                                                                                   ⁢                    π                    ⁢                                                                                   ⁢                    z                                    λ                                                                    ⅈ              ⁢                                                           ⁢              λ              ⁢                                                           ⁢              z                                ⁢                      ⅇ                          ⅈ              ⁢                              π                                  λ                  ⁢                                                                           ⁢                  z                                            ⁢                              (                                                      x                    2                                    +                                      y                    2                                                  )                                              ⁢                                                    ∫                ∫                            ∞                        ⁡                          [                                                                    U                    0                                    ⁡                                      (                                          ξ                      ,                      η                                        )                                                  ⁢                                  ⅇ                                      ⅈ                    ⁢                                          π                                              λ                        ⁢                                                                                                   ⁢                        z                                                              ⁢                                          (                                                                        ξ                          2                                                +                                                  η                          2                                                                    )                                                                                  ]                                ⁢                      ⅇ                                          -                ⅈ                            ⁢                                                2                  ⁢                  π                                                  λ                  ⁢                                                                           ⁢                  z                                            ⁢                              (                                                      x                    ⁢                                                                                   ⁢                    ξ                                    +                                      y                    ⁢                                                                                   ⁢                    η                                                  )                                              ⁢                      ⅆ            ξ                    ⁢                      ⅆ            η                                              (                  Equation          ⁢                                           ⁢          3                )            Basically, it is the Fourier transform of the field amplitude that is first multiplied by a quadratic phase (chirp). The expression for Fresnel diffraction can be rearranged to give                               U          ⁡                      (                          x              ,              y                        )                          =                                            ⅇ                              ⅈ                ⁢                                                      2                    ⁢                    π                    ⁢                                                                                   ⁢                    z                                    λ                                                                    ⅈλ              ⁢                                                           ⁢              z                                ⁢                                    ∫              ∫                        ∞                    ⁢                                    U              0                        ⁡                          (                              ξ                ,                η                            )                                ⁢                      ⅇ                          ⅈ              ⁢                                                π                                      λ                    ⁢                                                                                   ⁢                    z                                                  ⁡                                  [                                                                                    (                                                  x                          -                          ξ                                                )                                            2                                        +                                                                  (                                                  y                          +                          η                                                )                                            2                                                        ]                                                              ⁢                      ⅆ            ξ                    ⁢                      ⅆ            η                                              (                  Equation          ⁢                                           ⁢          4                )            which is simply the convolution of the field amplitude with a chirp. Fraunhofer diffraction occurs when the spherical wave can be approximated with a plane wave, which is a valid approximation if                     z        >>                              2            ⁢                          π              ⁡                              (                                                      x                    2                                    +                                      y                    2                                                  )                                                          2            ⁢                                                   ⁢            λ                                              (                  Equation          ⁢                                           ⁢          5                )            The field amplitude of U(x,y) after propagating a distance z can now be written as                               U          ⁡                      (                          x              ,              y                        )                          =                                            ⅇ                              ⅈ                ⁢                                                      2                    ⁢                                                                                   ⁢                    π                    ⁢                                                                                   ⁢                    z                                    λ                                                                    ⅈ              ⁢                                                           ⁢              λ              ⁢                                                           ⁢              z                                ⁢                      ⅇ                          ⅈ              ⁢                              π                                  λ                  ⁢                                                                           ⁢                  z                                            ⁢                              (                                                      x                    2                                    +                                      y                    2                                                  )                                              ⁢                                    ∫              ∫                        ∞                    ⁢                                    U              0                        ⁡                          (                              ξ                ,                η                            )                                ⁢                      ⅇ                                          -                ⅈ                            ⁢                                                2                  ⁢                  π                                                  λ                  ⁢                                                                           ⁢                  z                                            ⁢                              (                                                      x                    ⁢                                                                                   ⁢                    ξ                                    +                                      y                    ⁢                                                                                   ⁢                    η                                                  )                                              ⁢                      ⅆ            ξ                    ⁢                      ⅆ            η                                              (                  Equation          ⁢                                           ⁢          6                )            Basically this is simply the Fourier transform of the field amplitude, i.e.                                                                    U              ⁡                              (                                  x                  ,                  y                                )                                      =                                                            ⅇ                                      ⅈ                    ⁢                                                                  2                        ⁢                        π                        ⁢                                                                                                   ⁢                        z                                            λ                                                                                        ⅈ                  ⁢                                                                           ⁢                  λ                  ⁢                                                                           ⁢                  z                                            ⁢                              ⅇ                                  ⅈ                  ⁢                                      π                                          λ                      ⁢                                                                                           ⁢                      z                                                        ⁢                                      (                                                                  x                        2                                            +                                              y                        2                                                              )                                                              ⁢              F              ⁢                                                           ⁢              T              ⁢                              {                                                      U                    0                                    ⁡                                      (                                          ξ                      ,                      η                                        )                                                  }                                                                                    ξ            =                          x                              λ                ⁢                                                                   ⁢                z                                              ,                      η            =                          y                              λ                ⁢                                                                   ⁢                z                                                                        (                  Equation          ⁢                                           ⁢          7                )            If the optical aperture is lens of focal length f with a pupil function P(x,y), where                               P          ⁡                      (                          x              ,              y                        )                          =                  {                                                                      1                  ⁢                                                                           ⁢                                      if                    ⁡                                          (                                              x                        ,                        y                                            )                                                        ⁢                                                                                                     ⁢                                                                                                   ⁢                  is                  ⁢                                                                                                     ⁢                                                                                                   ⁢                  within                  ⁢                                                                           ⁢                  the                  ⁢                                                                           ⁢                  aperture                                                                                                      0                  ⁢                                                                           ⁢                                      if                    ⁡                                          (                                              x                        ,                        y                                            )                                                        ⁢                                                                                                     ⁢                                                                                                   ⁢                  is                  ⁢                                                                                                     ⁢                                                                                                   ⁢                  outside                  ⁢                                                                           ⁢                  the                  ⁢                                                                           ⁢                  aperture                                                                                        (                  Equation          ⁢                                           ⁢          8                )            then the amplitude distribution U0(x,y) behind the lens becomes                                           U            0                    ⁡                      (                          x              ,              y                        )                          =                                            U              i                        ⁡                          (                              x                ,                y                            )                                ⁢                      P            ⁡                          (                              x                ,                y                            )                                ⁢                      ⅇ                                          -                ⅈ                            ⁢                              π                                  λ                  ⁢                                                                           ⁢                  f                                            ⁢                              (                                                      x                    2                                    +                                      y                    2                                                  )                                                                        (                  Equation          ⁢                                           ⁢          9                )            where Ui(x,y) is the amplitude distribution in front of the lens. If Ui(x,y) is a plane wave and P(x,y) is a circular aperture, then for a properly focused diffraction-limited incoherent imaging system with a clear circular aperture, the optics PSF (point spread function) at the focal plane is given by substituting Equation 9 into Equation 7, resulting in the Airy function                                           PSF                          circular              ⁢                                                           ⁢              aperture                                ⁡                      (            r            )                          =                                                                                            U                                      circular                    ⁢                                                                                   ⁢                    aperture                                                  ⁡                                  (                                      x                    ,                    y                                    )                                                                    2                    =                                                                      (                                                            π                      ⁢                                                                                           ⁢                                              D                        2                                                                                    4                      ⁢                                                                                           ⁢                      λ                      ⁢                                                                                           ⁢                      f                                                        )                                2                            ⁡                              [                                                      2                    ⁢                                                                                   ⁢                                                                  J                        1                                            ⁡                                              (                                                                              π                            ⁢                                                                                                                   ⁢                            D                            ⁢                                                                                                                   ⁢                            r                                                                                λ                            ⁢                                                                                                                   ⁢                            f                                                                          )                                                                                                                        π                      ⁢                                                                                           ⁢                      D                      ⁢                                                                                           ⁢                      r                                                              λ                      ⁢                                                                                           ⁢                      f                                                                      ]                                      2                                              (                  Equation          ⁢                                           ⁢          10                )            where       r    =                            x          2                +                  y          2                      ,D is the diameter of the circular aperture, f is the focal length, λ is the wavelength of light, and J1® is the first-order Bessel function. The width of the PSF is generally defined by the diameter within the first zero, which is given by                                           PSF                          circular              ⁢                                                           ⁢              aperture                                ⁢          width                =                              2.44            ⁢                                          λ                ⁢                                                                   ⁢                f                            D                                =                      2.44            ⁢                                                   ⁢                          λ              ⁡                              (                                  f                  ⁢                  #                                )                                                                        (                  Equation          ⁢                                           ⁢          11                )            where f#≡f/D. The Rayleigh criterion for resolution is defined as the separation between two point sources such that the peak of one source is on the first zero of the second, which occurs at 1.22 λ(f#). The OTF (optical transfer function) of a diffraction-limited incoherent imaging system is the Fourier transform of the optics PSF, and the magnitude of the OTF is the MTF (modulation transfer function) of the optics. The optics MTF for an incoherent diffraction-limited optical system is essentially the aperture MTF, which is calculated by autocorrelating the aperture function. For a clear circular aperture of diameter D, the incoherent aperture MTF is given by                                           MTF            incoherent                    ⁡                      (            ρ            )                          =                                                            2                π                            ⁡                              [                                                                            cos                                              -                        1                                                              ⁡                                          (                                              ρ                        n                                            )                                                        -                                                            ρ                      n                                        ⁢                                                                  1                        -                                                  ρ                          n                          2                                                                                                                    ]                                      ⁢                                                   ⁢            for            ⁢                                                   ⁢            0                    ≤                      ρ            n                    ≤          1                                    (                  Equation          ⁢                                           ⁢          12                )                                                      MTF            incoherent                    ⁡                      (            ρ            )                          =                                   ⁢                              for            ⁢                                                   ⁢                          ρ              n                                >          1                                    (                  E          ⁢                                           ⁢          q          ⁢                                           ⁢          u          ⁢                                           ⁢          a          ⁢                                           ⁢          t          ⁢                                           ⁢          i          ⁢                                           ⁢          o          ⁢                                           ⁢          n          ⁢                                           ⁢          13                )                                where        ⁢                                                                                                  ρ          n                =                  ρ                      ρ            c                                              (                  E          ⁢                                           ⁢          q          ⁢                                           ⁢          u          ⁢                                           ⁢          a          ⁢                                           ⁢          t          ⁢                                           ⁢          i          ⁢                                           ⁢          o          ⁢                                           ⁢          n          ⁢                                           ⁢          14                )                                          ρ                      c            ⁡                          (              incoherent              )                                      =                              1                          λ              ⁡                              (                                  f                  ⁢                                                                           ⁢                  #                                )                                              =                      D                          λ              ⁢                                                           ⁢              f                                                          (                  E          ⁢                                           ⁢          q          ⁢                                           ⁢          u          ⁢                                           ⁢          a          ⁢                                           ⁢          t          ⁢                                           ⁢          i          ⁢                                           ⁢          o          ⁢                                           ⁢          n          ⁢                                           ⁢          15                )            and ρ is the radial spatial frequency. For coherent imaging systems with circular apertures, the MTF is simply the aperture function                                           MTF            coherent                    ⁡                      (            ρ            )                          =                              1            ⁢                                                                       ⁢                                                                     ⁢            for            ⁢                                                   ⁢            0                    ≤                      ρ            n                    ≤          1                                    (                  Equation          ⁢                                           ⁢          16                )                                          …          ⁢                                           ⁢                                    MTF              coherent                        ⁡                          (              ρ              )                                      =                              0            ⁢                                                   ⁢            for            ⁢                                                   ⁢                          ρ              n                                >          1                                    (                  Equation          ⁢                                           ⁢          17                )                                where        ⁢                                                                                                  ρ                      c            ⁡                          (              coherent              )                                      =                              1                          2              ⁢                                                           ⁢                              λ                ⁡                                  (                                      f                    ⁢                                                                                   ⁢                    #                                    )                                                              =                      D                          2              ⁢              λ              ⁢                                                           ⁢              f                                                          (                  Equation          ⁢                                           ⁢          18                )            Note that for both coherent and incoherent imaging systems there is a distinct spatial frequency cutoff, ρc, which is proportional to the aperture size and defines the highest spatial resolution that can be imaged with the optical system. An imaging system with a larger aperture size, therefore, will capture images at higher resolution than an imaging system with a smaller aperture size.
Sparse apertures (also called dilute apertures) use a reduced aperture area to synthesize the optical performance of a filled aperture. A sparse aperture system can combine the light captured by smaller apertures to capture a higher spatial resolution than possible from any of the individual apertures. This concept is very appealing in technology areas where a filled aperture is too large or heavy for the intended application. Sparse aperture concepts have been used to design large astronomical telescopes as well as small endoscopic probes (see U.S. Pat. No. 5,919,128 titled “SPARSE APERTURE ENDOSCOPE,” issued Jul. 6, 1999 to Fitch).
FIG. 1a illustrates a traditional Cassegrain telescope 10. FIG. 1b illustrates a prior art sparse aperture telescope 12 created by removing parts of the primary mirror of the Cassegrain telescope 10 in FIG. 1a. FIG. 1c illustrates a prior art sparse aperture telescope 14 created by using multiple afocal telescopes 16 that relay light into a combiner telescope 18 using an optical relay system 20 to precisely ensure that the light from each telescope arrives at a detector 21 simultaneously. These optical relay systems require many optical elements to move quickly and precisely to properly combine the light from each of the apertures. The number of these optical elements increases with the number of apertures, thus this alignment may prove too complex for imaging systems with many apertures.
Referring to FIG. 2, consider an image capture system comprising of three apertures 22, each with a diameter d and separated by a distance s. The diffraction-limited resolution of each aperture 22 is proportional to the diameter d, but if the electromagnetic wavefront from each aperture 22 is coherently combined, then a higher resolution can be captured as if collected by a single aperture 24 of diameter d+s. This requires the electromagnetic wavefront from each aperture 22 to be properly phased and coherently combined to form a high-resolution image.
To better understand the phasing of the individual wavefronts, first consider a single aperture imaging system shown in FIG. 3. An electromagnetic wavefront 28 from a scene 26 passes through an aperture 30 with lens a distance z1 from the scene 26 and results in the electromagnetic wavefront 32 after the aperture 30, described by Equation 9. The electromagnetic wavefront 32 can be described more generally as a wave function with amplitude a(x,y,z) and phase φ(x,y,z). An image I(x,y) captured by image sensor 34 a distance z2 from the aperture 30 only represents the intensity of the electromagnetic wavefront 28, given by                               I          ⁡                      (                          x              ,              y              ,                              z                2                                      )                          =                                                                                            a                  ⁡                                      (                                          x                      ,                      y                      ,                                              z                        2                                                              )                                                  ⁢                                  ⅇ                                                            -                      ⅈ                                        ⁢                                                                                   ⁢                                          ϕ                      ⁡                                              (                                                  x                          ,                          y                          ,                                                      z                            2                                                                          )                                                                                                                                2                    =                                    a              2                        ⁡                          (                              x                ,                y                ,                                  z                  2                                            )                                                          (                  Equation          ⁢                                           ⁢          19                )            
If the aperture 30 is replaced with N multiple smaller apertures then the wavefront from each aperture must be properly combined to maintain the resolution such that they coherently sum to form an image                               I          ⁢                      (                          x              ,              y              ,                              z                2                                      )                          =                                                                        ∑                                  n                  =                  1                                N                            ⁢                                                                    a                    n                                    ⁡                                      (                                          x                      ,                      y                      ,                                              z                        2                                                              )                                                  ⁢                                  ⅇ                                                            -                      ⅈ                                        ⁢                                                                                   ⁢                                                                  ϕ                        n                                            ⁡                                              (                                                  x                          ,                          y                          ,                                                      z                            2                                                                          )                                                                                                                                      2                                    (                  Equation          ⁢                                           ⁢          20                )            
If each aperture is a lens with focal length f and aperture P(x,y), then the intensity distribution of the image will be                               I          ⁡                      (                          x              ,              y                        )                          =                                                                        ∑                                  n                  =                  1                                N                            ⁢                                                                    U                    n                                    ⁡                                      (                                          x                      ,                      y                                        )                                                  ⁢                                  P                  ⁡                                      (                                          x                      ,                      y                                        )                                                  ⁢                                  ⅇ                                                            -                      ⅈ                                        ⁢                                                                                   ⁢                                          π                                              λ                        ⁢                                                                                                   ⁢                        f                                                              ⁢                                          (                                                                        x                          2                                                +                                                  y                          2                                                                    )                                                                                                                2                                    (                  Equation          ⁢                                           ⁢          21                )            where Un(x,y) is the amplitude distribution in front of the nth lens. Note that simply imaging the wavefronts from each aperture will not generate a high-resolution image because the electromagnetic wavefronts are not properly added; only the intensity values are added.
FIG. 4 illustrates a multiple aperture system, similar to the sparse aperture telescope 14 that combines the individual wavefronts as disclosed in U.S. Pat. No. 5,905,591 titled “MULTI-APERTURE IMAGING SYSTEM,” issued May 18, 1999 to Duncan et al. An electromagnetic wavefront 28 from a scene 26 passes through the multiple apertures 22 to produce wavefronts 36 that pass through an optical relay system 20, which coherently sums the wavefronts 36 from each aperture 22 with a set of movable mirrors 38. The resulting wavefront 40 from the optical relay system 20 is imaged by the combiner 42, which produces the proper wavefront 44 to be imaged by the imaging sensor 34. If the wavefronts are not properly summed then the resolution corresponding to the synthesized aperture will not be achieved.
Some designers of synthetic aperture arrays have developed probes (see U.S. Pat. No. 4,950,880 titled “SYNTHETIC APERTURE OPTICAL IMAGING SYSTEM,” issued Aug. 21, 1990 to Hayner and U.S. Pat. No. 5,093,563 titled “ELECTRONICALLY PHASED DETECTOR ARRAYS FOR OPTICAL IMAGING,” issued Mar. 3, 1992 to Small et al.) where each aperture produces an electrical signals representing the amplitude and phase of the incoming light and have the advantage that large-aperture resolutions are synthesized without optical phase compensating components. However, because each aperture in these approaches outputs a single electrical signal, requiring the need to acquire many images to build up the samples in Fourier space in order to form a two-dimensional image. An example of this approach is the Very Large Array (VLA) radio telescope, which uses a horn antenna at the prime focus to collect amplitude and phase information from each aperture and then combines the signals in a correlator to form the high-resolution image.
A holographic technique for generating high-resolution telescope images has been developed (see U.S. Pat. No. 5,283,672 titled “HOLOGRAPHIC TECHNIQUES FOR GENERATING HIGH RESOLUTION TELESCOPIC IMAGES,” issued Feb. 1, 1994 to Hong et al.) where a small lens is shifted laterally and a sequence of exposures is recorded on holographic medium. The plurality of holograms is illuminated with a coherent reference beam to reconstruct a high-resolution image that corresponds to a lens as large as the distance that the small lens was shifted. The advantage of this technique is that an image with the resolution corresponding to a large lens can be generated using a small lens. However, this technique requires a sequence of holograms and a coherent illumination source to illuminate the scene and reconstruct the image. Also, this technique does not use an image processor but relies on optical methods to generate the final image.
There is a need, therefore, for an image capture method that utilizes an image processor to coherently combine the electromagnetic wavefront from each aperture without the need for complex optical relay systems, a multitude of collections to build up the spatial frequencies of the scene, or a sequence of holograms.